change of order of integration

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1: Bibliography L

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Changing the order of integration. J. Austral. Math. Soc. 11, pp. 421–432.

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Notes:

An addendum appears in Vol. 14, pp. 383-384, of the same journal.

External Links:

Document, MathReview (G. W. Johnson), ZentralBlatt

Referenced by:

§1.5(v).

Encodings:

BibTeX

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Addendum to: “Changing the order of integration”. J. Austral. Math. Soc. 14, pp. 383–384.

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External Links:

Document, MathReview (G. W. Johnson), ZentralBlatt

Referenced by:

§1.5(v).

Encodings:

BibTeX

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2: 1.5 Calculus of Two or More Variables

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Change of Order of Integration

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3: 19.28 Integrals of Elliptic Integrals

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4: 2.6 Distributional Methods

… ►Motivated by Watson’s lemma (§2.3(ii)), we substitute (2.6.2) in (2.6.1), and integrate term by term. …Inserting (2.6.2) into (2.6.1) and integrating formally term-by-term, we obtain … ►The Stieltjes transform of $f(t)$ is defined by … ►On substituting (2.6.15) into (2.6.26) and interchanging the order of integration, the right-hand side of (2.6.26) becomes … ►In terms of the convolution product …

5: 1.17 Integral and Series Representations of the Dirac Delta

… ►Formal interchange of the order of integration in the Fourier integral formula ((1.14.1) and (1.14.4)): … ►In the language of physics and applied mathematics, these equations indicate the normalizations chosen for these non-$L^2$ improper eigenfunctions of the differential operators (with derivatives respect to spatial co-ordinates) which generate them; the normalizations (1.17.12_1) and (1.17.12_2) are explicitly derived in Friedman (1990, Ch. 4), the others follow similarly. … ►Formal interchange of the order of summation and integration in the Fourier summation formula ((1.8.3) and (1.8.4)): …

6: 2.10 Sums and Sequences

… ►Assume that $a,m$, and $n$ are integers such that $n>a$, $m>0$, and $f^{(2m)}(x)$ is absolutely integrable over $[a,n]$. Then … ►This identity can be used to find asymptotic approximations for large $n$ when the factor $v_j$ changes slowly with $j$, and $u_j$ is oscillatory; compare the approximation of Fourier integrals by integration by parts in §2.3(i). … ►

On the circle $|z|=r$, the function $f(z)-g(z)$ has a finite number of singularities, and at each singularity $z_j$, say,

2.10.30 $$f(z)-g(z)=O\left({(z-z_j)}^{{\sigma }_j-1}\right),$$ $z\to z_j$,

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Symbols:

$O\left(x\right)$: order not exceeding, $f(x)$: analytic function, $g(z)$: comparison function and ${\sigma }_j$: positive constant

Permalink:

http://dlmf.nist.gov/2.10.E30

Encodings:

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See also:

Annotations for §2.10(iv), §2.10 and Ch.2

where ${\sigma }_j$ is a positive constant.

… ►In these circumstances the integrals in (2.10.28) are integrable by parts $m$ times, yielding …

7: Errata

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8: Philip J. Davis

… ►An interesting anecdote told by Davis reveals that he and mathematician Philip Rabinowitz were dubbed “Heroes of the SEAC” when their Gaussian integration code executed correctly on its first run. … ►Davis also co-authored a second Chapter, “Numerical Interpolation, Differentiation, and Integration” with Ivan Polonsky. … ►Olver had been recruited to write the Chapter “Bessel Functions of Integer Order” for A&S by Milton Abramowitz, who passed away suddenly in 1958. … ►The surface color map can be changed from height-based to phase-based for complex valued functions, and density plots can be generated through strategic scaling. …

9: 2.5 Mellin Transform Methods

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§2.5(i) Introduction

… ►We now apply (2.5.5) with , and then translate the integration contour to the right. … ►Let $f(t)$ and $h(t)$ be locally integrable on $(0,\mathrm{\infty })$ and …Also, let … ►(The last order estimate follows from the Riemann–Lebesgue lemma, §1.8(i).) …

10: 2.3 Integrals of a Real Variable

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§2.3(i) Integration by Parts

… ►Then the series obtained by substituting (2.3.7) into (2.3.1) and integrating formally term by term yields an asymptotic expansion: … ►derives from the neighborhood of the minimum of $p(t)$ in the integration range. … ►A uniform approximation can be constructed by quadratic change of integration variable: … ►We replace the limit $\kappa$ by $\mathrm{\infty }$ and integrate term-by-term: …